The map φ(x,y)=(\sqrt{1+x²}-y,x) of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability: The plane is a union of smooth, disjoint, invariant curves of the map φ. However, the integral has not explicitly been known. In the current paper we will show that the map φ does not have an algebraic integral, i.e., there is no non-constant function F(x,y) such that:

• F∘φ=F.
• There exists a polynomial G(x,y,z) of three variables with G(x,y,F(x,y))=0.

Thus, the integral of φ, if it does exist, will have complicated singularities. We also argue that if there is an analytic integral F, then there would be a dense set of its level curves which are algebraic, and an uncountable and dense set of its level curves which are not algebraic.

This research has been supported in part by the National Science Foundation under grant no. DMS 9404419